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We show in a certain Lie*-algebra, the connections between the Lie subalgebra G:= G + G* + [ G, G*], generated by a Lie subalgebra G, and the properties of G. This allows us to investigate some useful information about the structure of such two Lie subalgebras. Some results on the relations between the two Lie subalgebras are obtained. As an application, we get the following conclusion: Let A ⊂ B( X) be a space of self-adjoint operators and ℒ:= A ⊕ i A the corresponding complex Lie*-algebra. G = G + G* + [ G, G*] and G are two LM-decomposable Lie subalgebras of ℒ with the decomposition G = R( G) + S, G = R + S, and R ⊆ R( G). Then G is ideally finite iff R:= R + R* + [ R, R] is a quasisolvable Lie subalgebra, S:= S + S + [ S, S] is an ideally finite semisimple Lie subalgebra, and [ R, S] = [ R, S] = {0}. [ABSTRACT FROM AUTHOR]

Abstract: Let denote the algebra of operators on a Hilbert . Let and denote the elementary operators ΔAB(X)=AXB−X and E(X)=AXB−CXD. We answer two questions posed by Turnšek [Mh. Math. 132 (2001) 349–354] to prove that: (i) if A, B are contractions, then if and only if is closed for some integer n⩾1; (ii) if A, B, C and D are normal operators such that A commutes with C and B commutes with D, then if and only if 0∈isoσ(E). [Copyright &y& Elsevier]